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1.107. (a) If x is a rational number whose square is less than 2, show that x 2 x2 =10 is a larger such number. (b) If x is a rational number whose square is greater than 2, nd in terms of x a smaller rational number whose square is greater than 2. 1.108. Illustrate how you would use Dedekind cuts to de nep p p p p p p p (a) 5 3; b 3 2; c 3 2 ; d 2= 3.
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Sequences
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DEFINITION OF A SEQUENCE A sequence is a set of numbers u1 ; u2 ; u3 ; . . . in a de nite order of arrangement (i.e., a correspondence with the natural numbers) and formed according to a de nite rule. Each number in the sequence is called a term; un is called the nth term. The sequence is called nite or in nite according as there are or are not a nite number of terms. The sequence u1 ; u2 ; u3 ; . . . is also designated brie y by fun g.
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EXAMPLES. 1. The set of numbers 2; 7; 12; 17; . . . ; 32 is a nite sequence; the nth term is given by un 2 5 n 1 5n 3, n 1; 2; . . . ; 7. 2. The set of numbers 1; 1=3; 1=5; 1=7; . . . is an in nite sequence with nth term un 1= 2n 1 , n 1; 2; 3; . . . .
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Unless otherwise speci ed, we shall consider in nite sequences only.
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LIMIT OF A SEQUENCE A number l is called the limit of an in nite sequence u1 ; u2 ; u3 ; . . . if for any positive number  we can nd a positive number N depending on  such that jun lj <  for all integers n > N. In such case we write lim un l.
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EXAMPLE . If un 3 1=n 3n 1 =n, the sequence is 4; 7=2; 10=3; . . . and we can show that lim un 3.
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If the limit of a sequence exists, the sequence is called convergent; otherwise, it is called divergent. A sequence can converge to only one limit, i.e., if a limit exists, it is unique. See Problem 2.8. A more intuitive but unrigorous way of expressing this concept of limit is to say that a sequence u1 ; u2 ; u3 ; . . . has a limit l if the successive terms get closer and closer to l. This is often used to provide a guess as to the value of the limit, after which the de nition is applied to see if the guess is really correct.
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THEOREMS ON LIMITS OF SEQUENCES If lim an A and lim bn B, then
n!1 n!1
1. 2. 3.
n!1 n!1 n!1
lim an bn lim an lim bn A B
n!1 n!1 n!1 n!1
lim an bn lim an lim bn A B lim an bn lim an lim bn AB
n!1 n!1
24 lim an n!1 an A n!1 bn lim bn B lim
SEQUENCES
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if lim bn B 6 0
a If B 0 and A 6 0, lim n does not exist. n!1 bn a If B 0 and A 0, lim n may or may not exist. n!1 bn 5. 6.
n!1 p lim an lim an p A p , n!1
for p any real number if A p exists.
lim pan p n!1 pA ,
liman
for p any real number if pA exists.
INFINITY We write lim an 1 if for each positive number M we can nd a positive number N (depending on n!1 M) such that an > M for all n > N. Similarly, we write lim an 1 if for each positive number M we can nd a positive number N such that an < M for all n > N. It should be emphasized that 1 and 1 are not numbers and the sequences are not convergent. The terminology employed merely indicates that the sequences diverge in a certain manner. That is, no matter how large a number in absolute value that one chooses there is an n such that the absolute value of an is greater than that quantity.
BOUNDED, MONOTONIC SEQUENCES If un @ M for n 1; 2; 3; . . . ; where M is a constant (independent of n), we say that the sequence fun g is bounded above and M is called an upper bound. If un A m, the sequence is bounded below and m is called a lower bound. If m @ un @ M the sequence is called bounded. Often this is indicated by jun j @ P. Every convergent sequence is bounded, but the converse is not necessarily true. If un 1 A un the sequence is called monotonic increasing; if un 1 > un it is called strictly increasing. Similarly, if un 1 @ un the sequence is called monotonic decreasing, while if un 1 < un it is strictly decreasing.
EXAMPLES. 1. The sequence 1; 1:1; 1:11; 1:111; . . . is bounded and monotonic increasing. It is also strictly increasing. 2. The sequence 1; 1; 1; 1; 1; . . . is bounded but not monotonic increasing or decreasing. 3. The sequence 1; 1:5; 2; 2:5; 3; . . . is monotonic decreasing and not bounded. However, it is bounded above.
The following theorem is fundamental and is related to the Bolzano Weierstrass theorem ( 1, Page 6) which is proved in Problem 2.23. Theorem. Every bounded monotonic (increasing or decreasing) sequence has a limit.
LEAST UPPER BOUND AND GREATEST LOWER BOUND OF A SEQUENCE A number M is called the least upper bound (l.u.b.) of the sequence fun g if un @ M, n 1; 2; 3; . . . while at least one term is greater than M  for any  > 0. " " A number m is called the greatest lower bound (g.l.b.) of the sequence fun g if un A m, n 1; 2; 3; . . . " while at least one term is less than m  for any  > 0. Compare with the de nition of l.u.b. and g.l.b. for sets of numbers in general (see Page 6).
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